To solve for the approximate length of the pole $p$ in the diagram, we use the three-dimensional Pythagorean theorem. This involves calculating the diagonal of a rectangular prism.

Step 1: Understand the setup

The flower bed is a rectangular prism with:

• Length = $8 \, \text{ft}$,
• Width = $6 \, \text{ft}$,
• Height = $3 \, \text{ft}$.

The pole $p$ extends diagonally from one corner at the base to the opposite corner at the top of the prism.

Step 2: Use the 3D Pythagorean Theorem

The formula for the length of a diagonal ($p$) in a rectangular prism is: $p = \sqrt{L^2 + W^2 + H^2}$ where $L$, $W$, and $H$ are the length, width, and height of the prism.

Step 3: Substitute values

$p = \sqrt{8^2 + 6^2 + 3^2}$ $p = \sqrt{64 + 36 + 9}$ $p = \sqrt{109}$

Step 4: Approximate the square root

$p \approx 10.4 \, \text{ft}$

Final Answer:

The approximate length of the pole is 10.4 feet to the nearest tenth.

Would you like a breakdown of the calculations or further clarifications?

Related Questions:

1. How do you calculate the diagonal length of a 2D rectangle?
2. How is the 3D Pythagorean theorem applied in real-life scenarios?
3. What are some other uses of the diagonal formula in construction or design?
4. How would the length change if the height of the flower bed increased by 1 foot?
5. How can you calculate the area of the flower bed's base?

Tip:

For any right-angled 3D problem, visualize breaking it down into two steps: first calculate the base diagonal, then use that diagonal with the height to find the 3D diagonal.